# Sample Paper for MA / M.Sc Mathematics of Complex Analysis !!

**Sample Paper for MA/M.Sc Mathematics of Complex Analysis**

**TERM END EXAMINATIONS**

**M. A. /M. Sc. – MATHEMATICS, YEAR – 1**

**COMPLEX ANALYSIS
**

#### Duration -3 Hours Max Marks: 60

**Note: 1. Attempt any FIVE questions. **

** 2. All questions carry equal marks.**

1. (a) Show that function : f(z) = (z ¹ 0), f(0) = 0 is not analytic at z = 0, although C.R. equations satisfied at that point.

(b) Derive the formula for the radius of convergence of the power series a_{n }z^{n} f find the same for the series.

2. (a Using Cauchy’s integral formula, prove that where C is the circle |z| = 3

(b) State & prove weiestrass theorem concerning the behaviour of an analytic function near an isolatial essential singularity.

3. (a) State & prove Argument principle

(b) Prove that every polynomial of degree n has no zeros & determine the number of roots of the equation

z^{8} – 4z^{5} + z^{2} – 1 = 0 that lies inside the circle |z| = 1

4. (a) State & prove minimum modulus principle.

(b) Show that cot z =

5. (a) State & prove Taylor’s Theorem.

(b) Expand f(z) = in Lament’s series valid for the regions

| < |z+1| < 2

6. (a) State & prove Fundamental theorem of Algebra with the help of Morera theorem.

(b) Expand log |1+z| is Taylor’s series about z = 0 & determine the region of convergence for the resulting series.

7. (a) There cannot be two different direct analytic continuations of a function

(b) If f(z) is memorphic inside a closed contour c & has no zero on c then

= N-P

N is the number of zeros & P the number of poles inside C.

8. (a) State & prove Rouche’s theorem,

(b) State & prove Mittag-Leffler Theorem.

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